In recent years several papers have been devoted to stability
and smoothing properties in maximum-norm of
finite element discretizations of parabolic problems.
Using the theory of analytic semigroups it has been possible
to rephrase such properties as bounds for the resolvent
of the associated discrete elliptic operator. In all these
cases the triangulations of the spatial domain has been
assumed to be quasiuniform. In the present paper we
show a resolvent estimate, in one and two space dimensions,
under weaker conditions on the triangulations than quasiuniformity.
In the two-dimensional case, the bound for the resolvent contains
a logarithmic factor.

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